Two inequalities between cardinal invariants

We prove two ZFC inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of ω of asymptotic density 0. We obtain an upper bound on the ∗-covering number, sometimes also called the weak covering number, of this ideal by proving that cov∗(Z0) k d. Next, we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove that, in sharp contrast to the case when k = ω, if k is any regular uncountable cardinal, then sk≤bk.